The composition of a L^2 function and a diffeomorphism is well-defined?

Let $f \in L^2((0,1);\mathbb{R})$ with respect to the Lebesgue measure.
Let $g$ be a $C^1$-diffeomorphism from $(0,2)$ into its image $g((0,2)) \subset (0,1)$.
Can I define the composition function $h:(0,2) \rightarrow \mathbb{R}$ by
$$h(x)=f(g(x)),$$
for almost every $x \in (0,2)$ ?
Is it enough to say that the measure of $g((0,2))$ is not zero ?